The present exemplary embodiment of the invention provides a new approach for generation of three dimensional fractal subsurface structures by Voronoi tessellation and computation of the gravity response of such fractal structure. This provides an efficient and new way of fractal subsurface generation, which removes possibilities of getting into a reentrant structure while perturbing Voronoi centers during iteration steps (involving in inverse modeling of the underlying structure and computation of the gravity response of such fractal subsurface, which constitutes forward modeling of the underlying structure for exploration of hydrocarbons and minerals).
It is known from gravitational law that bodies having mass exert attractional force on each other. In geophysical aspects instead of mass, the density plays a direct role in gravity surveys since density varies laterally as well as vertically, and hence affects the mass of a volume. It could be posed in a way that if a large object is having density contrast with its surroundings then it will reflect its signature in the observed gravity field. In case of geophysical studies the density contrast between different interfaces is responsible for the gravity anomaly, which in turn can be studied either for hydrocarbon exploration or for geological studies.
Exploration for hydrocarbons and minerals in subsurface environments with anomalous density variations have always been problems for traditional seismic imaging techniques by concealing geological structures beneath zones of anomalous density. Many methods for delineating the extent of the highly anomalous density zones exist. Exploration for hydrocarbons in subsurface environments with anomalous density variations such as salt formations, shale diapers create difficulties in seismic imaging techniques by concealing geologic structures beneath zones of anomalous density. By utilizing gravity, magnetic measurements and geological constraints along with a robust inversion process, these anomalous density zones can be modeled. The spatial resolution obtained from this process is normally much lower than that obtained from reflection seismic data. However, models obtained from gravity and magnetic data can provide a more accurate starting model for the seismic processing. Using the potential field data models as a starting model for two dimensional and three dimensional imaging by any technique greatly enhances the probability of mapping these concealed geologic structures beneath the zones of anomalous density. One source of geologic exploration data that has not been used extensively in spite of being cheapest in the past is potential fields data, such as gravity and magnetic data, and using potential fields data in combination with seismic data to provide a more accurate depth model which is a key parameter in any exploration program.
In a paper in Geophysical Journal of Royal Astronomical Society, v.3, 1960, Bott suggested a method to trace the floor of a sedimentary basin, which involved the approximation of a sedimentary basin by a series of two dimensional juxtaposed rectangular/square blocks of uniform density. The assumption of uniform density in entire rectangular/square blocks in said method is highly simplified case of actual reality. Computation of gravity anomalies due to two dimensional and three dimensional bodies of arbitrary shape has been given in Journal of Geophysical Research, v.64, 1959 by Talwani et al. and in Geophysics, v.25, 1960 by Talwani and Ewing which uses several variables in terms of co-ordinates of the corners of the polygons. The method is as accurate as much the number of corners is assumed to approximate the irregular shaped body, which enhances number of variables to be used in computation. The problem becomes much more complicated and cumbersome to deal with so many parameters when the said method is used in an inversion algorithm wherein the co-ordinates of the polygon corners are perturbed in every iteration in order to fit and achieve the best model. Yet another problem in said method arises when used in automated inversion algorithm is that of getting into a reentrant structure while perturbation of the polygon corner co-ordinates. This method became very popular and is widely used even in present days because of its mathematical simplicity. All these theories mentioned above, assumed subsurface structure consisting of some simple geometrical shape but Dimri opined the concept of fractal subsurface structure in chapter 16 “Fractal dimension analysis of soil for flow studies” pp. 189-193 In: Application of fractals in earth sciences, 2000, edited, V. P. Dimri, A. A. Balkema, USA. Benoit Mandelbrot, who coined the term fractal in his book “The fractal geometry of nature”, W. H. Freeman, New York, 1982 are put forth as the fractals are generally irregular (not smooth) in shape, and thus are not objects definable by traditional geometry. That means fractals tend to have significant detail, visible at any arbitrary scale; when there is self-similarity, this can occur because zooming in simply shows similar pictures, alternatively “A geometrical or physical structure having an irregular or fragmented shape at all scales of measurement” is known as fractal structure. Earlier in the Journal of Geophysical Research v. 100, 1995 Maus and Dimri had opined new concept of scaling behavior of potential fields and established a relation between the scaling exponent of source and field, which is useful for understanding of fractal geology. The fractal structure can be generated using very few parameters by Voronoi tessellation.
In a straight forward iterative algorithm for the planar Voronoi diagram, Information Process Letters, v. 34, 1990, Tipper illustrated that the Voronoi tessellation in a two dimensional space consists of enclosing every center by a Voronoi polygon (FIG. 1) such that common edge of adjacent polygons is perpendicular bisector to the line joining the centers on each side of that edge. Tipper illustrated Voronoi tessellation which uses least square distance formula i.e. L2 norm which limits the possibility and hence efficiency of generating variety of structures merely by changing the exponent p in the Lp norm. Here we have generalised the notion of Voronoi tessellation by using Lp distances instead of the least square distances as mentioned in ‘Deconvolution and Inverse Theory: Application to Geophysical Problems’, Elsevier Publication, 1992, by Dimri, V. P., so that Voronoi domains are not necessarily of polygonal shape (FIG. 2(b)).
In yet another study published in proceeding of Indian Academy of Sciences (Earth and Planetary Sciences) v. 108, 1999, Moharir et al. have advocated the use of economical sub-surface representational techniques for non-local optimization, wherein authors have used leminiscate representation which does not provide such an efficient and wider possibility of generating subsurface structure as given in the present invention. In the present invention a fractal sub-surface is generated which is close to the natural settings and domains of different physical property are assigned different colors, which can be later perturbed iteratively for inverse modeling, which is important in the interpretation of geophysical data as mentioned in the book by Dimri, V. P., 1992, Elsevier Science Publisher, Amsterdam.
There is a need for a method for efficient and accurate delineation of subsurface structure which is close to the real geology present below the earth surface in two dimensions and three dimensions. Such a method should preferably be able to use physical property variations in the subsurface geometry. Also, the method should preferably be able to obtain a realistic forward model of anomalous formations that are expected in a normal sedimentary basin, which can be used in exploration. The present invention satisfies this need.